متطلبات الإكمال
📌 Exercice 1
Écrire les ensembles suivants en extension :
a) \( A = \{x \mid x \text{ est un nombre entier impair entre 1 et 15}\} \)
b) \( B = \{x \mid x \text{ est un jour de la semaine comportant un a}\} \)
c) \( C = \{x \mid x \text{ est un nombre entier et } 2 < x < 8\} \)
✅ Corrigé
a) \( A = \{1, 3, 5, 7, 9, 11, 13, 15\} \)
b) \( B = \{\text{lundi, mardi, mercredi, jeudi, samedi}\} \)
c) \( C = \{3, 4, 5, 6, 7\} \)
📌 Exercice 2
Écrire les ensembles suivants en compréhension :
a) \( A = \{2, 4, 6, 8, 10\} \)
b) \( B = \{22, 23, 24, 25, 26, 27\} \)
c) \( C = \{a, e, i, o, u, y\} \)
✅ Corrigé
a) \( A = \{2k \mid k \in \mathbb{N}, 1 \leq k \leq 5\} \)
b) \( B = \{x \in \mathbb{N} \mid 22 \leq x \leq 27\} \)
c) \( C = \{x \mid x \text{ est une voyelle de l'alphabet français}\} \)
📌 Exercice 3
Soit \( E = \{\text{Anne, Bernard, Claude, Daniel, Émile, Fernand, Gaston}\} \).
a) \( A = \{x \in E \mid x \text{ comporte six lettres}\} \)
b) \( B = \{x \in E \mid x \text{ se termine par un e}\} \)
c) \( C = \{x \in E \mid x \text{ comporte un nombre impair de lettres}\} \)
d) \( D = \{x \in E \mid x \text{ commence par un l}\} \)
✅ Corrigé
a) \( A = \{\text{Anne (4 lettres)}, \text{Claude (6 lettres)}, \text{Fernand (7 lettres)}\} \) → \( A = \{\text{Claude, Fernand}\} \)
b) \( B = \{\text{Anne, Claude, Émile}\} \)
c) \( C = \{\text{Anne (4), Bernard (7), Claude (6), Daniel (6), Émile (5), Fernand (7), Gaston (6)}\} \) → \( C = \{\text{Bernard, Émile, Fernand}\} \)
d) \( D = \emptyset \) (aucun ne commence par l)
📌 Exercice 4
Soient \( A = \{0,1,2,3,4,5,6,7\}, B = \{1,3,5,6\}, C = \{2,3,4,7\}, D = \{1\} \).
Compléter par \( \in, \notin, \subset \text{ ou } \subsetneq \) :
- a) 1 ___ A
- b) 7 ___ B
- c) 5 ___ B
- d) {3} ___ C
- e) B ___ A
- f) D ___ C
- g) 4 ___ C
- h) {1} ___ D
✅ Corrigé
a) \( 1 \in A \)
b) \( 7 \notin B \)
c) \( 5 \in B \)
d) \( \{3\} \subset C \) (ou \( \{3\} \subsetneq C \) car \( C \) a plus d'éléments)
e) \( B \subset A \) (ou \( B \subsetneq A \))
f) \( D \not\subset C \) (car \( 1 \notin C \))
g) \( 4 \in C \)
h) \( \{1\} = D \) donc \( \{1\} \subset D \) et \( D \subset \{1\} \)
📌 Exercice 6
Soient \( A = \{1,3,5,7,9\}, B = \{0,2,4,6,8\}, C = \{2,3,4\}, D = \{1,3,7,8\} \).
Déterminer :
- a) \( A \cup C \)
- b) \( A \cap B \)
- c) \( C \cap D \)
- d) \( C \cup D \)
✅ Corrigé
a) \( A \cup C = \{1,2,3,4,5,7,9\} \)
b) \( A \cap B = \emptyset \)
c) \( C \cap D = \{3\} \)
d) \( C \cup D = \{1,2,3,4,7,8\} \)
📌 Exercice 7
Soient \( A = \{1,2,3,5,8\} \) et \( B = \{x \in \mathbb{N}^* \mid x \text{ est pair et } x < 11\} \).
a) Déterminer \( A \cup B \) et \( A \cap B \).
b) Faire le diagramme de Venn.
✅ Corrigé
\( B = \{2,4,6,8,10\} \).
a) \( A \cup B = \{1,2,3,4,5,6,8,10\} \)
\( A \cap B = \{2,8\} \)
📌 Exercice 8
Soient \( A = \{2,3,5,7,11,13\}, B = \{3,6,9,12,15\}, C = \{2,3,8,9,13,15\} \).
a) Déterminer \( A \cap B, A \cap C \) et \( B \cap C \).
b) Faire le diagramme de Venn.
✅ Corrigé
a) \( A \cap B = \{3\} \)
\( A \cap C = \{2,3,13\} \)
\( B \cap C = \{3,9,15\} \)
📌 Exercice 9
Soit \( E = \{\text{Gaston, Paul, Jean, Pierre, Jeanne, Guy, Fernand, Georges}\} \).
\( A = \{x \in E \mid x \text{ commence par G}\} \), \( B = \{x \in E \mid x \text{ contient un a}\} \), \( C = \{x \in E \mid x \text{ comporte six lettres}\} \).
✅ Corrigé
\( A = \{\text{Gaston, Guy, Georges}\} \)
\( B = \{\text{Gaston, Jean, Jeanne, Fernand}\} \)
\( C = \{\text{Gaston, Pierre, Jeanne, Georges}\} \) (6 lettres)
\( A \cap B = \{\text{Gaston}\} \)
\( A \cap C = \{\text{Gaston, Georges}\} \)
\( B \cap C = \{\text{Gaston, Jeanne}\} \)
\( A \cup B = \{\text{Gaston, Guy, Georges, Jean, Jeanne, Fernand}\} \)
📌 Exercice 10
Recopier et compléter :
- a) \( \mathbb{N} \cap \mathbb{N}^* = \)
- b) \( \mathbb{N} \cap \emptyset = \)
- c) \( \mathbb{N} \cup \mathbb{N}^* = \)
- d) \( \mathbb{N}^* \cup \{0\} = \)
✅ Corrigé
a) \( \mathbb{N} \cap \mathbb{N}^* = \mathbb{N}^* \)
b) \( \mathbb{N} \cap \emptyset = \emptyset \)
c) \( \mathbb{N} \cup \mathbb{N}^* = \mathbb{N} \)
d) \( \mathbb{N}^* \cup \{0\} = \mathbb{N} \)
📌 Exercice 11
Soit A et B deux ensembles. Écrire en compréhension les ensembles \( A \cup B \) et \( A \cap B \).
✅ Corrigé
\( A \cup B = \{x \mid x \in A \text{ ou } x \in B\} \)
\( A \cap B = \{x \mid x \in A \text{ et } x \in B\} \)
📌 Exercice 12
Soient \( A = \{a, b, f, k, l, n, p\}, B = \{b, c, d, e, f\}, C = \{a, b\}, D = \{d, e, f, g\}, E = \{h, i, j, m, o\} \).
Déterminer les ensembles suivants :
- a) \( A \cap B \)
- b) \( B \cap D \)
- c) \( C \cap E \)
- d) \( A \cap D \)
- e) \( B \cap C \)
- f) \( A \cup B \)
- g) \( A \cup C \)
- h) \( A \cup B \cup C \cup D \cup E \)
✅ Corrigé
a) \( A \cap B = \{b, f\} \)
b) \( B \cap D = \{d, e, f\} \)
c) \( C \cap E = \emptyset \)
d) \( A \cap D = \{f\} \)
e) \( B \cap C = \{b\} \)
f) \( A \cup B = \{a, b, c, d, e, f, k, l, n, p\} \)
g) \( A \cup C = A = \{a, b, f, k, l, n, p\} \)
h) \( A \cup B \cup C \cup D \cup E = \{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p\} \)